The fate of the cosmos equation

[victorvmotti]

Will the cosmos expand forever or the expansion will stop and then the whole universe collapses to that point of spacetime from which the big bang started?

Carroll (2004) shows that this will become clear in a formula that shows the fate of the cosmos:

Ω0a^3+(1- ΩM0- Ω)a+ ΩM0=0

This simple cubic equation in terms of a, the scale factor, gives us predictive power. If there is no real solution to it then we have to expect “perpetual” expansion. And the current experimental data indeed favor such an open ended cosmic future.

What do you think about this conclusion?

 

[George Jones]

I am not sure what it is that you are asking.

Yes, observational evidence, when combined standard Friedmann-Lemaitre-Robertson-Walker models, strongly favours a universe that expands forever.

The same observation evidence and models, however, make no strong prediction about whether the universe is open or closed, i.e.., the universe seems to be very near the borderline.

 

[phinds]

Will the cosmos expand forever or the expansion will stop and then the whole universe collapses to that point of spacetime from which the big bang started?

It can't possibly collapse to the same point in spacetime where it started because the arrow of time only goes one way.

The "big crunch" scenario was in favor before accelerated expansion was discovered a couple of decades ago but is now widely regarded as not what's going to happen.

EDIT: I should add that the term "point in spacetime" should, I think, be taken to mean not just a particular point in time but also a particular point in space and since the big bang did not happen at a point in space, that's another reason why saying that it could go back to the "same" point in spacetime is impossible.

 

[Bill_K]

This simple cubic equation in terms of a, the scale factor, gives us predictive power. If there is no real solution to it then we have to expect “perpetual” expansion. And the current experimental data indeed favor such an open ended cosmic future.
What do you think about this conclusion?

I think that every cubic equation has a real solution.

 

[George Jones]

I think that every cubic equation has a real solution.

Carroll should have specified explicitly that turn-around values of the scale factor $a$ need to be real and positive, i.e., only positive solutions of the cubic are physically valid. He produces the real, positive solutions.

 

[victorvmotti]

Great feedback.

Yes, he begins by seeing when the Hubble parameter passes through zero, changes sign, from positive to negative, and then arrives at this cubic equation in terms of the scale factor.

I was wondering if this cubic equation captures all information and related values, here only density parameters of matter and vacuum, to predict the fate of the cosmos, either a crunch or a perpetual expansion.